EE 539B Integrated Optics From Micron Scale to Nanophotonics. 2.1 Waveguide input and output coupling 2.2 Coupling Between Waveguides

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1 EE 539B Integrated Optics From Micron Scale to Nanophotonics Waveguide Coupling.1 Waveguide input and output coupling. Coupling Between Waveguides Integrated Optics Theory and Technology, by R. G. Hunsperger, 5th ed., Ch. 7-8, Springer Verlag. EE 539B - 1

2 General Definitions for Coupling Loss x P in P m Coupling efficiency to the m-th mode η m = P P m in Q: A single mode optical beam is coupled into a waveguide with guiding core dimension a few times larger than the wavelength. What kind of modes will be generated in the waveguide? Coupling loss (db) L =10log P P in m EE 539B -

3 Direct Focusing η A( x, y): B m m = ( x, y): A( x, y) A( x, y) B * m dxdy ( x, y) dxdy Fielddistribution of B( x, y) Fielddistribution of dxdy theincident beam them- th mode In most cases, A(x,y) can be represented by Gaussian beams. EE 539B - 3

4 4 EE 539B - TEM 0,0 Gaussian Beam Radialphase ) ( ) ( exp alphase Longitudin tan exp factor Amplitude ) ( exp ) ( ) ( = R y x k j k j W y x W W A r A ) ( + = W W + = 0 1 ) ( R λ π 0 0 W Beam radius Ref: Verdeyen, Laser Electronics, 3 rd ed., Prentice Hall. Beam spreading Wavefront change Radius of curvature of the wavefront Rayleigh range Q: Which factor affects the coupling most?

5 Gaussian Beam Through a Thin Lens EE 539B - 5

6 6 EE 539B - End-Butt Coupling π + π + π + = η cos 1) ( 1 1 cos ) ( 1) ( 64 m t t t m t t t n n n n m L g L g L g g L g L m Approximation: (assuming all waveguide modes are well confined, and ) g t L t Exact coupling efficiency can be obtained by overlap integrals.

7 Misalignment Effect Lateral misalignment Longitudinal misalignment Q: Why is it oscillating? Can we eliminate the oscillation? P P 0 cos πx = tl for t g < t L, X t L t g Example MathCAD program for fiber-waveguide coupling. EE 539B - 7

8 Tapered Mode Sie Converters EE 539B - 8

9 Prism Couplers Air-waveguide coupling Phase-matching condition βm = kn1 sinθ m cannot be satisfied. Prism-waveguide coupling Phase-matching condition β m = knp sin θm can be satisfied. (Assuming normal incidence to the prism.) EE 539B - 9

10 Example: Output Prism Coupler A prism coupler with index n p =. is used to observe the modes of a waveguide. The light source is a He-Ne laser with λ 0 = 63.8 nm. If the light from a particular mode is seen at an angle of 6.43º with the normal to the prism surface, what is the propagation constant β m for that mode? Q: What is the interaction length required to obtain complete coupling? EE 539B - 10

11 Coupled-Mode Theory κ L = π Q: What will happen if κl > π/? κ : Coupling coefficient (depending on overlap integral between the prism mode and the waveguide mode) W L = = cosθ m π κ For a given L, the coupling coefficient required for complete coupling: L κ = πcos θ W m EE 539B - 11

12 Notes on Prism Coupling In order to get 100% coupling with a uniform beam, the trailing edge of the beam must exactly intersect the right-angle corner of the prism. Disadvantages For most semiconductor waveguides, β m ~ kn Difficult to find prism materials Incident beam must be highly collimated Coupling efficiency sensitive to the separation between the prism and the waveguide EE 539B - 1

13 Grating Coupler β 0 Periodic structure of the grating perturbs the waveguide modes in the region underneath the grating. ν π β ν =β 0 +, ν= 0, ± 1, ±,... Λ β0 : Propagation constant of the m-th mode covered by the grating β ~ β 0 m Phase-matching condition: β ν = kn1 sin θ m can be satisfied even though β m > kn1 EE 539B - 13

14 Example of Grating Coupler Grating: Λ = 0.4 μm on a GaAs planar waveguide λ 0 = 1.15 μm Propagation constant for the lowest-order mode in the waveguide: β 0 = 3.6k Assume 1 st -order coupling, ν = 1, what incident angle should the light make in order to couple to the lowest-order mode? At what λ 0 do we start to need higher-order coupling? EE 539B - 14

15 Directional Couplers Coupling: Mixing of two adjacent modes, exchanging power as they propagate along adjacent paths. Energy transfer in a coherent fashion. Direction of propagation maintained. EE 539B - 15

16 Synchronous Versus Asynchronous Synchronous: Both waveguides are identical Asynchronous: Both waveguides are not identical In-phase In-phase In-phase In-phase In-phase In-phase Out of phase In-phase Out of phase In-phase Power transfer continues all the time. Complete power transfer The power transfer that occurs while the waves are in phase is reversed when the waves are out of phase. Incomplete power transfer EE 539B - 16

17 Multilayer Planar Waveguide Coupler Ref: G. A. Vawter, J. L. Mer, and L. A. Coldren, Monolithically integrated transverse-junction-strip laser with an external waveguide in GaAs/AlGaAs, J. Quantum Electronics, v. 5, no., p , EE 539B - 17

18 Dual-Channel Directional Coupler Fraction of the power coupled per unit length determined by overlaps of the modes. Determine the amount of transmitted power by bending away the secondary channel at proper point. EE 539B - 18

19 Transmission Characteristics 3dB directional coupler, interaction length = 1mm 100% directional coupler, interaction length =.1mm EE 539B - 19

20 Coupled-Mode Theory Synchronous Coupling Electric field of the propagating mode in the waveguide: jβ1 jβ Exy (,, ) = E1( ) U1( xye, ) + E( ) U( xye, ) = A1( ) U1( x, y) + A( ) U( x, y) U ( xy, ) : Normalied field distribution in an unperturbed waveguide 1, Power flow in the waveguides: Coupled-mode equations: 1 Initial condition: da ( ) = jβa1( ) jκa( ) d da ( ) = jβa( ) jκa1( ) d κ : Coupling coefficient A1 (0) = 1 A (0) = 0 P ( ) = A ( ) 1, 1, Solutions: A1 ( ) = cos( κ)exp( jβ) A ( ) = jsin( κ)exp( jβ) β=βr j α α : Loss coefficient EE 539B - 0

21 Power Transfer in Synchronous Coupling Power flow: 1 1 P( ) = A( ) = cos ( κ)exp( α) P( ) = A ( ) = sin ( κ)exp( α) Phase in the driven guide always lag 90º behind the phase of the driving guide. Q: What s the consequence of this? Length necessary for complete transfer of power from one waveguide to the other. EE 539B - 1

22 Asynchronous Coupled-mode equations: 1 Define: Coupled-Mode Theory Asynchronous Coupling β1 β da ( ) = jβ1a1( ) jκ1a( ) d da ( ) = jβa( ) jκ1a1( ) d A() a () = exp( jβ ) 1 1 A() a() β : Coupled propagation constant Condition for non-trivial solutions results in: Initial condition: β=β± g β+β β g A (0) = 1 1 A (0) = 0 1 Δβ κ1κ 1 + EE 539B -

23 Power Transfer in Asynchronous Coupling Power flow: α Δβ 1( ) cos ( ) sin ( g) P = g e + e g κ1κ1 α P( ) = A1( ) = sin ( g) e g α Ψ= g (= κ for synchronous coupling) Assuming lossless for the figures. EE 539B - 3

24 What is κ, κ 1, and κ 1? Exact solution: ( ε ε) U * k0 1 c U1 da κ = β ( ε ε ) U U U * k0 c 1 1 U da κ = β da da Approximation: (For well-confined modes) κ= β W q + qs hqe ( h ) EE 539B - 4

25 Applications: Modulators and Switches Exercise: Let s design a modulator using directional coupler MathCAD program Control Δβ electrically 1 w x w y d n 1 n 0.8 n c P0( Δβ) P1( Δβ) Δβ 0.05 Q1: How to choose the waveguide length? Q: What is the best Δβ range? Q3: How to control Δβ electrically? Lose coupling EE 539B - 5

26 Directional Couplers as Modulators Synchronous coupling, κ 1 = κ 1 = Control Δβ electrically := 300 Coupling length, in μm 1 ( ) := κ1 κ1 g Δβ P0 Δβ + Δβ ( ) := cos( g( Δβ) ) + Δβ ( ( ) ) sin g Δβ g( Δβ) P0( Δβ) P1( Δβ) P1( Δβ) := κ1 κ1 g( Δβ) sin g Δβ ( ( ) ) Δβ 0.05 EE 539B - 6

27 Directional Couplers as Switches L = π κ κ : coupling coefficient Waveguide Synchronous coupling Cross state. When the effective indices, and therefore propagation constants, in the two waveguides are sufficiently different by applying bias, no coupling will occur Bar-state L Electrode := 300 Coupling length, in μm ( ) := κ1 κ1 g Δβ P0 Δβ + Δβ ( ) cos( g( Δβ) ) := + Δβ sin ( g ( Δβ ) ) g( Δβ) P0( Δβ) P1( Δβ) P1( Δβ) := κ1 κ1 g( Δβ) sin( g( Δβ) ) Δβ 0.3 EE 539B - 7

Optical Communications

Optical Communications Telecommunication Engineering School of Engineering University of Rome La Sapienza Rome, Italy 2005-2006 Lecture #2, May 2 2006 The Optical Communication System BLOCK DIAGRAM OF